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Research Papers

# The Estimates of the Mean First Exit Time of a Bistable System Excited by Poisson White Noise

[+] Author and Article Information
Yong Xu

Department of Applied Mathematics,
Northwestern Polytechnical University,
Xi'an 710072, China;
Potsdam Institute for Climate Impact Research,
Potsdam 14412, Germany;
Department of Physics,
Humboldt University of Berlin,
Berlin 12489, Germany
e-mail: hsux3@nwpu.edu.cn

Hua Li, Wantao Jia, Xiaole Yue

Department of Applied Mathematics,
Northwestern Polytechnical University,
Xi'an 710072, China

Haiyan Wang

School of Marine Sciences,
Northwestern Polytechnical University,
Xi'an 710072, China

Jürgen Kurths

Potsdam Institute for Climate Impact Research,
Potsdam 14412, Germany;
Department of Physics,
Humboldt University of Berlin,
Berlin 12489, Germany

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received February 27, 2017; final manuscript received June 19, 2017; published online July 12, 2017. Assoc. Editor: Walter Lacarbonara.

J. Appl. Mech 84(9), 091004 (Jul 12, 2017) (8 pages) Paper No: JAM-17-1111; doi: 10.1115/1.4037158 History: Received February 27, 2017; Revised June 19, 2017

## Abstract

We propose a method to find an approximate theoretical solution to the mean first exit time (MFET) of a one-dimensional bistable kinetic system subjected to additive Poisson white noise, by extending an earlier method used to solve stationary probability density function. Based on the Dynkin formula and the properties of Markov processes, the equation of the mean first exit time is obtained. It is an infinite-order partial differential equation that is rather difficult to solve theoretically. Hence, using the non-Gaussian property of Poisson white noise to truncate the infinite-order equation for the mean first exit time, the analytical solution to the mean first exit time is derived by combining perturbation techniques with Laplace integral method. Monte Carlo simulations for the bistable system are applied to verify the validity of our approximate theoretical solution, which shows a good agreement with the analytical results.

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## Figures

Fig. 1

The potential function V(x) for different symmetry parameter μ: (a) μ=0 and (b) μ=0.2

Fig. 2

The MFET+(xs1→xs2) in the symmetric potential when μ=0, *: Monte Carlo result, —: theoretical result. (a) E[Y2]=0.1, I0=0.4:0.01:0.8; (b) E[Y2]=0.001, I0=0.4:0.01:0.8; (c) E[Y2]=0.1, I0=1:3:28; (d) E[Y2]=0.01, I0=1:3:28; and (e) the difference between the Monte Carlo uMC+(−2) and the approximate theoretical solution u+(−2)=u0(−2)+ε2u2(−2) is plotted against ε2.

Fig. 3

The MFET+(xs1→xs2) in the asymmetric potential when μ=0.2, *: Monte Carlo result, —: theoretical result. (a) E[Y2]=0.1, I0=0.26:0.03:0.8; (b) E[Y2]=0.001, I0=0.26:0.03:0.8; (c) E[Y2]=0.1, I0=1:3:28; (d) E[Y2]=0.01, I0=1:3:28; and (e) the difference between the Monte Carlo uMC+(xs1) and the approximate theoretical solution u+(xs1)=u0+(xs1)+ε2u2+(xs1) is plotted against ε2.

Fig. 4

The MFET(xs2→xs1) in the asymmetric potential when μ=0.2, *: Monte Carlo result, —: theoretical result. (a) E[Y2]=0.1, I0=0.6:0.05:2; (b) E[Y2]=0.001, I0=0.6:0.05:2; (c) E[Y2]=0.1, I0=1:3:28; (d) E[Y2]=0.01, I0=1:3:28; and (e) the difference between the Monte Carlo uMC−(xs2) and the approximate theoretical solution u−(xs2)=u0−(xs2)+ε2u2−(xs2) is plotted against ε2.

Fig. 5

The probability density p: (a) Poisson white noise WP with λ=2800, E[Y2]=0.01 and (b) Gaussian white noise WG with the same intensity

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